. Winslow's Comprehensive mathematics : being an extensive cabinet of numerical, arithmetical, and mathematical facts, tables, data, formulas, and practical . the altitude ofthe triangle A D C, therefore, A C being base. dTt ^-ac = oe, V (DC2 — Glf) = E D. ALT -f- A E = A C. D~C -r-CE = A C. ED2 -^AE = CE. DC : AC A C X AE= XT) CE = D~ E X C E = ET) : A~C2 :: A E :DC2 : AD9 : : C E ::: C E : A C. E. Twice the area of a right-angled triangle, divided by the hypo-tenuse, is equal the perpendicular to the hypotenuse, and therespective triangle will thereby be divided i
. Winslow's Comprehensive mathematics : being an extensive cabinet of numerical, arithmetical, and mathematical facts, tables, data, formulas, and practical . the altitude ofthe triangle A D C, therefore, A C being base. dTt ^-ac = oe, V (DC2 — Glf) = E D. ALT -f- A E = A C. D~C -r-CE = A C. ED2 -^AE = CE. DC : AC A C X AE= XT) CE = D~ E X C E = ET) : A~C2 :: A E :DC2 : AD9 : : C E ::: C E : A C. E. Twice the area of a right-angled triangle, divided by the hypo-tenuse, is equal the perpendicular to the hypotenuse, and therespective triangle will thereby be divided into two right-angledtriangles having a side common to both. LINES AND SUPERFICIES. 187 Half the product of the two legs of a right-angled triangle equalsthe area of that triangle. 2 area perp. to hypot. hypotenuse. 2 areagiven leg = required leg. Of Oblique-angled Triangles. Every triangle not a right-angled triangle is either an acute-angled triangle or an obtuse-angled triangle, and these two (theacute-angled and obtuse-angled) are classed under the general nameoblique-angled. The following principles are alike applicable toeither. Let ABC be the AO-BC 2AB r£AB = AD, distance along the base, fromthe angle formed by the baseand longest vertical side, atwhich a perpendicular droppedfrom the vertical angle will fall;KB and V(AC2 — AD2) =D C, theperpendicular alluded to ; thus dividing the obtuse-angled triangleABC into two right-angled triangles, A D C and B D C, D C a legcommon to both. Or, V (A C + A D X A C — A D) = D C ; for the sum of anytwo quantities multiplied by their difference is equal to the differ-ence of their squares. AB~ A CJ 2BC perpendicular to B C produced. -f h B C = B g, and a/(A B2 — B^2) = A ABJ-B C2 2 AC pendicular to A C produced. {- ^ = A h, and */( A B2 - A V) = B h, per- AC=AB+BC-2ABXBD j^ _ AB-fBC —AC2AB -2 2 2 188 LINES AND SUPERFICIES. 2 2 2 AC+AB—BC A J— 2p) Tc — ATff== Cg1an&Cg-\-BC = Bg;Ag +C#=AC. Twice the area of an
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